Nonparametric Estimation via Mixed Derivatives

了解一个特定的感兴趣的变量和一组相关的协变量之间的精确关系是科学和工程的许多领域中出现的一个基本问题。通常有必要将联合收割机两种不同的信息源结合起来来解决这个问题。信息的第一个来源是领域知识或理论，它通常为变量之间的关系提供某些基本形式。第二个信息来源是一组实验条件或受试者的变量值的观察数据。只有有效地结合这两种信息来源，才能有效地解决问题。制定这样一种方法是本项目的主要目标。假设控制变量之间关系的函数在某种意义上是光滑的，然后引入观测数据，以找到满足假设的光滑性的函数，该函数最好地解释了观测数据。该项目将探索执行该计划的不同方式，并开发对广泛科学领域的从业者有用的新方法和计算算法。大多数现有的研究这些问题的方法要么作出强的先验假设，这是不现实的，并导致错误的结论之间的关系的变量，或弱的先验假设，导致需要不切实际的巨大规模的数据集的可靠结论。基于“混合偏导数光滑约束”的方法有效地折衷了这两种极端的方法，并将导致具有很大实用价值的方法。从这项研究中产生的材料将通过研讨会和本科生以及研究生教学传播。将通过开发软件包，向更广泛的统计和科学界提供该项目的方法。本计画主要研究在平滑性限制下，包含混合偏导数的非参数函数估计问题。研究人员计划扩大最近开发的方法，非参数回归下的混合偏导数的第一和第二顺序，以允许限制相互作用的订单，设计更快的算法计算，并证明理论的准确性，更一般的设计假设下的结果。在强稀疏性设置下接近参数速率的可能性将在涉及张量积基（包括复指数和径向基函数）的更一般的设置中进行探索。不确定性量化将系统地探讨使用贝叶斯方法，主要集中在柯西先验。将在形状约束回归中探索新的混合导数方法，包括基于具有限制相互作用的整体单调性和总Popoviciu凸性的方法。该奖项反映了NSF的法定使命，并已被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Understanding the precise relationship between a specific variable of interest and a related set of covariables is a fundamental problem arising in many areas of science and engineering. It is usually necessary to combine two different sources of information to solve this problem. The first source of information is domain knowledge or theory, which often suggests certain basic forms for the relationship between the variables. The second source of information is observed data on the values of the variables for a set of experimental conditions or subjects. Effective solutions to the problem can only be obtained by efficiently combining both sources of information. The development of such a methodology is the primary goal of this project. Assuming the function governing the relationship between the variables is smooth in a certain sense, the observed data is then brought in to find the function satisfying the assumed smoothness that best explains the observed data. This project will explore different ways of carrying out this scheme and develop novel methods and computational algorithms useful to practitioners in broad scientific areas. Most existing methods for studying these problems either make strong prior assumptions that are unrealistic and lead to wrong conclusions on the relationship between the variables, or weak prior assumptions that lead to requiring unrealistically enormous sizes of datasets for reliable conclusions. The developed approach based on "mixed partial derivative smoothness constraints" effectively compromises these two extreme approaches and will lead to methods of great practical value. The material emanating from this research will be disseminated through seminars and undergraduate as well as graduate teaching. The methods from this project will be made available to the wider statistics and scientific community through the development of software packages. This project focuses on nonparametric function estimation problems under smoothness constraints involving mixed partial derivatives. The investigator plans to expand recently developed methodology for nonparametric regression under mixed partial derivatives of first and second orders to allow for restricted interaction orders, design faster algorithms for computation, and prove theoretical accuracy results under more general design assumptions. The possibility of near parametric rates under strong sparsity settings will be explored in a more general setting involving tensor product bases (including complex exponentials and radial basis functions). Uncertainty quantification will be systematically explored using Bayesian approaches with a main focus on Cauchy priors. New mixed derivative approaches will be explored in shape-constrained regression, including those based on Entire Monotonicity with restricted interactions and total Popoviciu convexity. Mixed derivative approaches for spectral density estimation of time series and density estimation will also be studied.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.